We explain the proof of Theorem REF . Due to the lack of velocity dissipation in two
directions, we take the functional setting to be the Sobolev space \(H^3\)  in order to guarantee
the uniqueness. The local existence follows from a standard procedure (see, e.g., [1]}), so
we focus on the global a priori bounds of \((u, b)\) . This is accomplished via the bootstrapping
argument (see, e.g., [2]}). A crucial step is to construct a suitable energy
functional. Naturally it should include the \(H^3\) -norm together with the time integral pieces
from the dissipative terms
\(E^{(1)}_0(t)=\sup _{0\le \tau \le t}\Vert (u(\tau ),b(\tau ))\Vert _{H^3}^2+\int _0^t \Big (\Vert \partial _1 u(\tau )\Vert _{H^3}^2+\Vert \nabla _h b(\tau )\Vert _{H^3}^2\Big )d \tau .\) 
